AI 中文总结
本文证明在简单阿贝尔簇幂中，当两条不可约代数曲线不满足特定子群包含条件时，满足倍数映射落入另一曲线的点集（除去使整条曲线包含的倍数）是有限的。
AI 中文摘要
设$G$是定义在$\mathbb{Q}^\mathrm{alg}$上的$g \in \mathbb{N}$维简单阿贝尔簇，且$C_1, C_2 \subseteq G^N(\mathbb{C})$是$N \geq 3$时的不可约闭代数曲线。进一步假设$C_1$和$C_2$中至少有一个不在$\mathbb{Q}^\mathrm{alg}$上定义。假设不存在$G^N(\mathbb{C})$中维度为$g$的代数子群$G$使得$C_1 \subseteq G$，且不存在$G^N(\mathbb{C})$中维度为$2g$的代数子群$H$使得$C_1 \cup C_2 \subseteq H$。记$\mathcal{N} = \{n \in \mathbb{N} \\ | \\ [n]C_1 \subseteq C_2\}$，我们证明$\bigcup_{n \in \mathbb{N} \setminus \mathcal{N}}\{x \in C_1 \\ | \\ x^n \in C_2\}$是有限的。
英文摘要
Let $G$ be a simple abelian variety of dimension $g \in \mathbb{N}$ defined over $\mathbb{Q}^\mathrm{alg}$ and let $C_1, C_2 \subseteq G^N(\mathbb{C})$ be irreducible closed algebraic curves with $N \geq 3$. Further assume that at least one of $C_1$ and $C_2$ is not defined over $\mathbb{Q}^\mathrm{alg}$. Suppose that there does not exist an algebraic subgroup $H_1 \subseteq G^N(\mathbb{C})$ of dimension $g$ such that $C_1 \subseteq H_1$ and that there does not exist an algebraic subgroup $H_2 \subseteq G^N(\mathbb{C})$ of dimension $2g$ such that $C_1 \cup C_2 \subseteq H_2$. Denoting $\mathcal{N} = \{n \in \mathbb{N} \ | \ [n]C_1 \subseteq C_2\}$, we prove that $\bigcup_{n \in \mathbb{N} \setminus \mathcal{N}}\{x \in C_1 \ | \ nx \in C_2\}$ is finite.
